+1 on the multi-pump, like a Daisy 880 or similar. I have the others, springers, CO2, rifles and pistols, but the multi-pump is the one I go for at 3 am when there is a mouse or rat in the house or backyard. The reason is it is as accurate as anything else, is easy to shoot accurately and I can adjust the power! Nobody wants an uneccessary hole in the wall.
Somebody on another channel was asking about those Gamo PBA pellets. That might be a good in-home choice as they don't ricochet as much as a BB, but won't freak out mom if a toddler eats one.
Outside the house, unless you have a _lot_ of mice and rats, sniping one here and there with the multipump is easy.